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Efficient Nonnegative Matrix Factorization by DC Programming and DCA.

Hoai An Le Thi1, Xuan Thanh Vo2, Tao Pham Dinh3

  • 1Department for Management of Science and Technology Development and Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam, and Laboratory of Theoretical and Applied Computer Science EA 3097, University of Lorraine, Ile du Saulcy, 57045 Metz, France lethihoaian@tdt.edu.vn.

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This study introduces two novel approaches using difference of convex functions (DC) programming and the DC algorithm (DCA) to solve nonnegative matrix factorization (NMF) and its variants. These methods demonstrate competitive efficiency against state-of-the-art algorithms.

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Area of Science:

  • Optimization
  • Machine Learning
  • Numerical Analysis

Background:

  • Nonnegative Matrix Factorization (NMF) is a widely used dimensionality reduction technique.
  • Existing NMF algorithms often face challenges with convergence and efficiency for various NMF variants.
  • The difference of convex functions (DC) programming and DC algorithm (DCA) offer a powerful framework for solving non-convex optimization problems.

Purpose of the Study:

  • To develop novel DC programming and DCA-based algorithms for solving the NMF problem and its numerous variants.
  • To analyze the convergence properties of the proposed algorithms.
  • To demonstrate the efficiency and competitiveness of the proposed methods compared to existing state-of-the-art algorithms.

Main Methods:

  • Two main approaches based on DC programming and DCA are developed.
  • The first approach utilizes an alternating framework solving non-negativity constrained least squares subproblems via DCA.
  • The second approach directly applies DCA to the entire NMF problem, with variations including variable selection.

Main Results:

  • The proposed algorithms are adapted to solve a wide range of NMF variants, including sparse, smooth, multilayer, convex, and symmetric NMF.
  • It is shown that the developed algorithms encompass several existing methods for NMF variants as special cases.
  • Empirical results on real-world and synthetic datasets show favorable performance compared to five state-of-the-art alternating nonnegative least squares algorithms.

Conclusions:

  • The DC programming and DCA-based approaches provide a unified and efficient framework for solving NMF and its variants.
  • The proposed algorithms offer competitive performance and include existing methods as special cases, highlighting their generality.
  • These methods represent a significant advancement in solving challenging NMF problems.